Real eigenvalues and nonzero eigenvectors of 2×2 matrices

Mathematical statement

For a 2×2 real matrix M, a real λ is an eigenvalue exactly when some nonzero v∈R² satisfies Mv=λv. This pulse uses the nonzero-eigenvector convention. The candidate-scalar kernel Kλ={v:Mv=λv} includes zero even when λ is not an eigenvalue. Distinct real eigenvalues are the distinct real roots of λ²−tr(M)λ+det(M)=0.

Representative formula (exact TeX)

E(M)

Assumptions

• Work throughout over R with 2×2 matrices; not every real matrix has a real eigenvalue.
• Eigenpair-candidate questions permit a zero-vector input but must reject it. Kernel-membership questions permit and accept the zero vector.
• For Δ=tr(M)²−4det(M), negative, zero, and positive values give zero, one, and two distinct real eigenvalues respectively. A repeated root is not counted twice by algebraic multiplicity.

Symbol scope

• E — scoped real-eigenvalue summary
• M — 2×2 real square matrix

Intuition, definition, and misconceptions

The equation Mv=λv is automatic when v=0; it does not make every λ an eigenvalue. Keep the nonzero-eigenvector condition separate from membership in the kernel.

For diag(a,d), the distinct real-eigenvalue set is {a,d}, listing a=d only once. This does not establish multiplicities, an eigenbasis, or Jordan structure.

For a general real 2×2 matrix, compute the discriminant and number of distinct real roots exactly. A positive nonsquare discriminant still gives two distinct real roots, without floating approximations of those roots. General diagonalization and complex spectra are outside this pulse.

Common errors include accepting zero as an eigenvector, inferring an eigenvalue from zero in the kernel, or counting a double root as two distinct values.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Matrix-represented linear maps on the real plane — Use the linear model Mv.
• Determinants of real 2×2 matrices — Supply the constant coefficient of the characteristic polynomial.
• Quadratic equations over the reals — Use distinct real roots and the discriminant slice.

Verification boundary

Real 2×2 integer instances can be recomputed; general spectral, diagonalization, multiplicity, and eigenspace theory have no attached formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind pinned Module.End.HasEigenvector and eigenspace APIs to the nonzero condition and 2×2 characteristic polynomial, retaining the real-root domain.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.